Regularity of nonlinear generalized functions: a counterexample in the nonstandard setting

Regularity theory in generalized function algebras of Colombeau type is largely based on the notion of \({\mathcal G}^\infty\)-regularity, which reduces to \(C^\infty\)-regularity when restricted to Schwartz distributions. Surprisingly, in the nonstandard version of the Colombeau algebras, this basic property of \({\mathcal G}^\infty\)-regularity does not hold.